The ballistic limit of the log-Sobolev constant equals the Polyak-Łojasiewicz constant

The Polyak-Lojasiewicz (PL) constant of a function $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to \mathbb{R}$ characterizes the best exponential rate of convergence of gradient flow for $f$, uniformly over initializations. Meanwhile, in the theory of Markov diffusions, the log-Sobolev (LS) constant plays an analogous role, governing the exponential rate of convergence for the Langevin dynamics from arbitrary initialization in the Kullback-Leibler divergence. We establish a new connection between optimization and sampling by showing that the low temperature limit ${\lim }_{t\to 0^{+}}{t}^{-1}{C}_{\mathsf{L}\mathsf{S}}({\mu }_t)$ of the LS constant of ${\mu }_t\propto \exp (-f\mathrm{/}t)$ is exactly the PL constant of $f$, under mild assumptions. In contrast, we show that the corresponding limit for the Poincaré constant is the inverse of the smallest eigenvalue of ${\mathrm{\nabla }}^{2}f$ at the minimizer.